More answers

I understand the slope and that the line passes those points. However, how to get the formula and the piecewise function. I don't understand. Could you please clarify it for me?

jim_thompson5910

2 years ago

they broke the function into 3 linear pieces
each piece is really a full linear equation, but restricted on the domain

jim_thompson5910

2 years ago

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jim_thompson5910

2 years ago

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korosh23

2 years ago

ok

jim_thompson5910

2 years ago

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jim_thompson5910

2 years ago

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korosh23

2 years ago

one question

jim_thompson5910

2 years ago

go ahead

korosh23

2 years ago

for the second lienear function f(x)= -x +2
why 2 is not the y-intercept?

jim_thompson5910

2 years ago

extend out the middle piece in both directions
you'll see that piece cross 2 on the y axis

jim_thompson5910

2 years ago

|dw:1443244906871:dw|

korosh23

2 years ago

right :D

korosh23

2 years ago

one more thing, for the third function.
why don't we say
y=0 when x greater or equal to 2. Because at 2, y is 0

jim_thompson5910

2 years ago

notice the \(\Large 1 < x \le 2\) as the restriction on the second piece
if x = 2, then you use y = 2-x to find y
if we used \(\Large x \ge 2\) for the third piece, then that would mean x = 2 is defined for the third piece as well. Question is: which piece do we pick if x = 2? This ambiguity is a bad thing when it comes to functions. So that's why the restrictions are set up in a way to avoid letting x = 2 be defined for more than one piece

jim_thompson5910

2 years ago

put another way, the fancy notation you see at the bottom really breaks down into this
f(x) = x if 0 <= x <= 1
OR
f(x) = 2-x if 1 < x <= 2
OR
f(x) = 0 if x > 2
each piece can't overlap another piece. Otherwise, we wouldn't have a function

korosh23

2 years ago

what do you mean by overlap?

korosh23

2 years ago

I believe connected does not mean overlapped

jim_thompson5910

2 years ago

by overlap, I mean defined in multiple places

korosh23

2 years ago

Oh I see, and the last question. Can we say we have 3 formulas for the function?

DanJS

2 years ago

Is the formal way to put the solid point on the left function, or can you do whatever for the intervals

korosh23

2 years ago

Ok thank you @DanJS

jim_thompson5910

2 years ago

think of the 3 restrictions as number lines|dw:1443245579767:dw|

DanJS

2 years ago

oh i didnt see tha tin the book , i was just wondering

jim_thompson5910

2 years ago

now if we had x >= 2 for the third piece, then we have this overlap
|dw:1443245694279:dw|

korosh23

2 years ago

Right, hollow ones mean the number itself does not include

korosh23

2 years ago

So if both are full. means overlapped becaue the numbers are present in two different equations.

jim_thompson5910

2 years ago

exactly

korosh23

2 years ago

@jim_thompson5910 thank you my friend. I felt like I was in a classroom. lol

DanJS

2 years ago

is it legal to reverse each solid - dot combo in between functions?

DanJS

2 years ago

**
solid - open

DanJS

2 years ago

i am guessing yes, since it describes the same thing, maybe

jim_thompson5910

2 years ago

`is it legal to reverse each solid - dot combo in between functions?`
I'm not sure what you mean. You mean use "solid" to mean "open/excluded endpoint" ? If so, then you're going against all of the conventional notation I've seen. That would be like making "up" be "down"

DanJS

2 years ago

I think that step function wasopen - solid

DanJS

2 years ago

typo corrected in next line, i just mean can you specify the intervals another way, for each interface between functions, can you choose arbitrarily which function will include the value

jim_thompson5910

2 years ago

oh, yeah you could say maybe
0 <= x < 1
1 <= x < 2
x >= 2
just as long as there is no overlap, you're fine

DanJS

2 years ago

k, that is what i was trying to say, solid-open endpoint combo switch deal.. blahh